Abstract:
The thesis comprises of generalized inequalities for monotone functions from which
we deduce important inequalities such as reversed Hardy type inequalities, general-
ized Hermite-Hadamard’s inequalities etc by putting suitable functions. The present
thesis is divided into three chapters.
The first chapter includes generalized inequalities given for C-monotone functions
and multidimensional monotone functions. As a result of these inequalities, we de-
duce reversed Hardy inequalities for C-monotone functions and multidimensional re-
versed Hardy type inequalities with the optimal constant. Furthermore, we construct
functionals from the differences of above inequalities and gives their n-exponential
convexity and exponential convexity. By using log-convexity of these functionals we
give refinements of these inequalities. Also we give mean-value theorems for these
functionals and deduce Cauchy means for them.
The second chapter consists of inequalities valid for monotone functions of the form
f /h and f /h. These are also very interesting as by putting suitable functions we
get one side of Hermite-Hadamard’s inequality and generalized Hermite-Hadamard’s
inequality. Similarly as in the first chapter, we make functionals of these inequalities
and gives results regarding n-exponential convexity and exponential convexity. Also
we give mean value theorems of Lagrange and Cauchy type as well as we obtain non-
symmetric Stolarsky means with and without parameter.
In the third and the last chapter we consider Petrovi ́ type functionals obtained from
c
Petrovi ́ type inequalities and investigate their properties like superadditivity, sub-
c
additivity, monotonicity and n-exponential convexity.
Also at the end of each chapter we discuss examples in which we construct further
exponential convex functions and their relative properties.